Hodge dual definition

Question

I'm reading Christodoulou and Klainerman's book on the stability of Minkowski spacetime and came upon the definition of the Hodge dual on a surface $S$: for a 2-form $\xi$, for instance, $$ *\xi_{ab} = \epsilon_{ac}\xi^c_b. $$ But I've usually seen the Hodge dual as something that sends 2-forms on a surface to scalars, and in general (co)tensors of rank $k$ to (co)tensors of rank $n - k$, where $n$ is the dimension of the manifold. Are they two related? Is there a reason for this discrepancy?